The present invention relates generally to three dimensional (3D) computerized tomography (CT) and more specifically to methods for constructing and assessing a three dimensional scanning trajectory to ensure the acquisition of a complete set of Radon data for exact image reconstruction of an object irradiated by a cone beam source.
Conventional CT employs a technique for obtaining cross sectional slices of an object from planar parallel or fan beam irradiation of an object. The technique is primarily utilized in medical and industrial diagnostics. Traditional image reconstruction techniques have been predominantly two dimensional. In three dimensions, an undistorted image of an object can be mathematically reconstructed in an exact manner by back projecting a parallel beam which has been attenuated after passing through the object using an inverse transform based on the Fourier Slice Theorem. The use of a parallel beam source and a flat two dimensional detector geometrically simplifies reconstruction but complicates practical considerations having to do with speed and ease of data collection.
Back projections can be mathematically accomplished for a 3D cone beam source by inverse Radon transforming suitable planar integrals. The planar integrals are computed from detector integrals which utilize measured cone beam projection data i.e. the detected attenuated intensity representative of the density distributions of the irradiated object. The use of a 3D cone beam source expedites data acquisition, but complicates geometrical considerations when used with a flat detector.
In two dimensions, the analog of cone beam source geometry is illustrated by fan beam geometry. For the case of fan beam geometry, the detector integral are equivalent to the Radon transform of the object. Unlike the two dimensional case, a direct Radon inversion of three dimensional cone beam data from a cone beam source is not possible. Before the inverse Radon transform can be undertaken in three dimensions, the cone beam detector integrals must be reconfigured into planar integrals suitable for inverse Radon transformation. Due to such limitations, three dimensional CT imaging has usually involved stacking slices representative of the density distribution through the object obtained from various parallel or fan beam attenuation projections. Each projection is associated with a particular view angle or configuration of source and detector relative to the object. A data set is generally acquired by either rotating a source and detector, fixed relative to each other, around an object taking projections as the object is scanned; or alternatively, rotating the object between the fixed source and detector.
The three dimensional Radon inversion problem was addressed in two commonly assigned patent applications: U.S. patent application Ser. No. 07/631,815 filed Dec. 18, 1990 by Kwok C. Tam entitled METHOD AND APPARATUS FOR CONVERTING CONE BEAM X-RAY PROJECTION DATA TO PLANAR INTEGRAL AND RECONSTRUCTING A THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT discloses method and apparatus for converting cone beam data to values representing planar integrals on any arbitrary set of planes in Radon space for 3D image reconstruction through inverse Radon transformation. A related U.S. patent application Ser. No. 07/631,818 filed on Dec. 21, 1990 by Kwok C. Tam entitled PARALLEL PROCESSING METHOD AND APPARATUS FOR RECONSTRUCTING THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY (CT) IMAGE OF AN OBJECT FROM CONE BEAM PROJECTION DATA OR FROM PLANAR INTEGRALS discloses a two step approach for performing an inverse Radon transform from planar integrals obtained on a plurality of coaxial planes. The first step involves calculating from the planar integrals a two dimensional projection image of the object on each of the coaxial planes; while the second step involves defining normal slices through these coaxial planes from which a two dimensional reconstruction of each slice is obtained. In this slice by slice way, the reconstruction algorithms operate on the plurality of planar integrals to produce a three dimensional image of the object.
It is further essential to note that the acquired data set is complete only if it provides sufficient Radon data at every necessary point in Radon space, i.e. Radon space must be sufficiently filled with data over the region of support in Radon space which corresponds to that region of support in object space occupied by the object. Radon data is generally acquired by exposing the entire object within the field of view of a source and scanning about the object using a source fixed with respect to a corresponding detector to obtain measurements. Sufficient filling of Radon space by a candidate scanning configuration is necessary for exact image reconstruction. Furthermore, if the detector integral space is filled over the region of support for the object, the Radon data set is complete. Bruce D. Smith in an article entitled "Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods," IEEE Trans. Med. Imag., MI-4 (1985) 14, has shown that a cone beam data set is complete if each plane passing through the object cuts the scanning trajectory in at least one point. This criterion assumes that the detector is fixed relative to the source and that the entire object can be scanned within the field of view of the source. Depending on the scanning configuration employed to obtain the cone beam projection data, the data set in Radon space may or may not be complete. Utilizing an incomplete data set for image reconstruction by Radon inversion introduces artifacts which compromise image quality and may render the image inadequate for medical or industrial diagnostic use.
A scanning configuration comprising two circular trajectories whose axes of rotation are normal with respect to one another is suggested by Gerald N. Minerbo, "Convolutional Reconstruction from Cone-Beam Projection Data", IEEE Trans. Nucl. Sci., Vol. NS-26, No. 2, pp. 2682-2684 (April 1979); and Heang K. Tuy, "An Inversion Formula for Cone-Beam Reconstruction", SIAM J. Math., Vol. 43, No. 3, pp. 546-552 (June 1983). Smith points out in his 1985 article that this trajectory satisfies the completeness criterion. Although complete, this scanning configuration is not practical as it is mechanically difficult to implement. A much easier to implement complete scanning trajectory has been disclosed in commonly assigned U.S. patent application Ser. No. 07/572,651, filed Aug. 27, 1990, by Eberhard et al entitled "SQUARE WAVE CONE BEAM SCANNING TRAJECTORY FOR DATA COMPLETENESS IN THREE-DIMENSIONAL COMPUTERIZED TOMOGRAPHY". Although incomplete, the scanning geometry most commonly adopted is the circular scanning trajectory which engulfs the entire object in the field of view of the source. A circular scanning configuration which minimizes data incompleteness by utilizing more than one circular scan path is disclosed in commonly-assigned U.S. patent application Ser. No. 07/572,590, filed Aug. 27, 1990, by Eberhard entitled "DUAL PARALLEL CONE BEAM CIRCULAR SCANNING TRAJECTORIES FOR REDUCED DATA INCOMPLETENESS IN THREE DIMENSIONAL COMPUTERIZED TOMOGRAPHY".
One can attempt to compensate for an inherently incomplete scanning trajectory using apriori corrections. Commonly assigned U.S. patent application Ser. No. 07/572,590 discloses an apriori approach to reducing the effects of incompleteness on three dimensional cone beam reconstruction by correcting two dimensional projection images obtained on each of a plurality of coaxial planes in Radon space using optically obtained object boundary information. From this, a three dimensional image is reconstructed on slices normal to the common axis in a slice by slice manner using two dimensional reconstruction on each slice.
Generally, no easy to implement, method for assessing whether a source scanning trajectory satisfies the completeness criterion has been disclosed which can accommodate exact image reconstruction in three dimensions when cone beam geometry is employed. Three dimensional exact image reconstruction is by its very nature computationally intensive; therefore, ensuring that data is collected in a manner that sufficiently fills the necessary volume of Radon space without unduly escalating the complexity of practical data acquisition provides a significant improvement over the existing art. As a practical consideration, efficient scanning also minimizes dose exposure to the object being scanned.
A typical scanning and data acquisition configuration employing cone beam geometry is depicted in FIG. 1. An object 20 is positioned within the field of view between a cone beam point source 22 and a typical two dimensional detector array 24, which provides cone beam projection data. An axis of rotation 26 passes through the field of view and the object 20. For purpose of analysis, a midplane 28 is defined normal to the axis of rotation 26 which contains the cone beam point source 22. By convention, the axis of rotation 26 is generally taken to be the z axis, having its origin at its intersection with the midplane. The (x,y,z) coordinate system is fixed relative to the source 22 and detector 24. In scanning the object 20 at a plurality of angular positions, the source 22 moves relative to the object and the field of view typically rotates along a circular scanning trajectory 30 lying in the midplane 28, while the detector 24 remains fixed with respect to the source 22 (or alternatively the object 20 can be rotated while the source 22 and detector 24 remain stationary). Data is acquired at a plurality of source positions during the scan. Data collected at the detector 24 represent line integrals through the object 20. The approach to reconstruction then embodies calculating planar integrals on a set of planes from various line integrals through the object, then performing an inverse Radon transform on these planar integrals to reconstruct a three dimensional image of the object.
It has already been established that data collected using a commonly adopted single circular scan is incomplete and artifacts may accordingly be introduced into the reconstructed image. Dual parallel circular scanning trajectories have been shown to reduce data set incompleteness. A circular square wave scanning trajectory, as well as, dual mutually perpendicular circular scanning trajectories provide a complete Radon data set for exact image reconstruction having been shown to satisfy the completeness criterion as articulated by Smith. More recently, Bruce D. Smith in an article entitled "Cone-beam Tomography: Recent Advances and a Tutorial Review", Optical Engineering, Vol. 29, No. 5, pp. 524-534, May 1990, mentions several complete scanning trajectories. However, Smith's statement of the completeness criterion which asserts that any plane through the object must intersect at least one point on the scanning trajectory is not easy to visualize nor practical to implement for candidate scanning trajectories.